The three-body problem in physics is recognized as a complex dynamic interaction between three bodies with mass. These bodies interact with each other through gravitational forces. While the principle of calculating their motion given initial conditions is known, even a slight change in one parameter can alter the dynamics so radically that the system almost becomes chaotic. In this paper, recognizing the potential of neural networks, we attempt to approach the problem from a new perspective. By generating initial conditions numerically through computational methods, data on the positions of the three bodies were collected, along with their velocities and mutual distances. The forces acting between them were also implemented in the code. Using the Runge-Kutta method, time steps were calculated, and the system's evolutions were saved in files suitable for Long Short-Term Memory (LSTM) recurrent neural networks. By analyzing various system evolutions, insights into the system and the laws governing its dynamics could be deduced. Predictions of the next step in the system's evolution were attempted. The Python programming language and a sufficiently powerful computer were used to create a suitable working environment. Neural networks require strong graphics cards and extensive calibration to obtain accurate solutions. From these results, it was concluded that these methods are promising. However, given the complexity of the system, a significant amount of data is needed for the network to learn effectively. With the resources available, quite good results were achieved. For further advancements, even more powerful computers, more data, and further calibration of settings will be necessary.